### All Complex Analysis Resources

## Example Questions

### Example Question #1 : Residue Theory

Cauchy's Residue Theorem is as follows:

Let be a simple closed contour, described positively. If a function is analytic inside except for a finite number of singular points inside , then

Brown, J. W., & Churchill, R. V. (2009). *Complex variables and applications*. Boston, MA: McGraw-Hill Higher Education.

Use Cauchy's Residue Theorem to evaluate the integral of

in the region .

**Possible Answers:**

**Correct answer:**

Note, for

a singularity exists where . Thus, since where is the only singularity for inside , we seek to evaluate the residue for .

Observe,

The coefficient of is .

Thus,

.

Therefore, by Cauchy's Residue Theorem,

Hence,

### Example Question #1 : Residue Theory

Cauchy's Residue Theorem is as follows:

Let be a simple closed contour, described positively. If a function is analytic inside except for a finite number of singular points inside , then

Brown, J. W., & Churchill, R. V. (2009). *Complex variables and applications*. Boston, MA: McGraw-Hill Higher Education.

Using Cauchy's Residue Theorem, evaluate the integral of

in the region

**Possible Answers:**

**Correct answer:**

Note, for

a singularity exists where . Thus, since where is the only singularity for inside , we seek to evaluate the residue for .

Observe,

The coefficient of is .

Thus,

.

Therefore, by Cauchy's Residue Theorem,

Hence,

### Example Question #3 : Residue Theory

Cauchy's Residue Theorem is as follows:

Let be a simple closed contour, described positively. If a function is analytic inside except for a finite number of singular points inside , then

Brown, J. W., & Churchill, R. V. (2009). *Complex variables and applications*. Boston, MA: McGraw-Hill Higher Education.

Use Cauchy's Residue Theorem to evaluate the integral of

in the region .

**Possible Answers:**

**Correct answer:**

Note, there is one singularity for where .

Let

Then

so

.

Therefore, there is one singularity for where . Hence, we seek to compute the residue for where

Observe,

So, when , .

Thus, the coefficient of is .

Therefore,

Hence, by Cauchy's Residue Theorem,

Therefore,

### Example Question #2 : Residue Theory

Cauchy's Residue Theorem is as follows:

For the following problem, use a modified version of the theorem which goes as follows:

Residue Theorem

If a function is analytic everywhere in the finite plane except for a finite number of singular points interior to a positively oriented simple closed contour , then

*Complex variables and applications*. Boston, MA: McGraw-Hill Higher Education.

Use the Residue Theorem to evaluate the integral of

in the region .

**Possible Answers:**

0

**Correct answer:**

0

Note,

Thus, seeking to apply the Residue Theorem above for inside , we evaluate the residue for .

Observe,

The coefficient of is .

Thus,

.

Therefore, by the Residue Theorem above,

Hence,

### Example Question #3 : Residue Theory

Cauchy's Residue Theorem is as follows:

For the following problem, use a modified version of the theorem which goes as follows:

Residue Theorem

If a function is analytic everywhere in the finite plane except for a finite number of singular points interior to a positively oriented simple closed contour , then

*Complex variables and applications*. Boston, MA: McGraw-Hill Higher Education.

Use the Residue Theorem to evaluate the integral of

in the region .

**Possible Answers:**

**Correct answer:**

Note,

Thus, seeking to apply the Residue Theorem above for inside , we evaluate the residue for .

Observe,

The coefficient of is .

Thus,

.

Therefore, by the Residue Theorem above,

Hence,

### Example Question #5 : Residue Theory

Cauchy's Residue Theorem is as follows:

For the following problem, use a modified version of the theorem which goes as follows:

Residue Theorem

If a function is analytic everywhere in the finite plane except for a finite number of singular points interior to a positively oriented simple closed contour , then

*Complex variables and applications*. Boston, MA: McGraw-Hill Higher Education.

Use the Residue Theorem to evaluate the integral of

in the region .

**Possible Answers:**

**Correct answer:**

Note,

Thus, seeking to apply the Residue Theorem above for inside , we evaluate the residue for .

Observe,

The coefficient of is .

Thus,

.

Therefore, by the Residue Theorem above,

Hence,

### Example Question #6 : Residue Theory

Cauchy's Residue Theorem is as follows:

For the following problem, use a modified version of the theorem which goes as follows:

Residue Theorem

*Complex variables and applications*. Boston, MA: McGraw-Hill Higher Education.

Use the Residue Theorem to evaluate the integral of

in the region .

**Possible Answers:**

**Correct answer:**

Note,

Thus, seeking to apply the Residue Theorem above for inside , we evaluate the residue for .

Observe, the coefficient of is .

Thus,

.

Therefore, by the Residue Theorem above,

Hence,

### Example Question #7 : Residue Theory

Cauchy's Residue Theorem is as follows:

For the following problem, use a modified version of the theorem which goes as follows:

Residue Theorem

*Complex variables and applications*. Boston, MA: McGraw-Hill Higher Education.

Use the Residue Theorem to evaluate the integral of

in the region .

**Possible Answers:**

**Correct answer:**

Note,

Thus, seeking to apply the Residue Theorem above for inside , we evaluate the residue for .

Observe,

The coefficient of is .

Thus,

.

Therefore, by the Residue Theorem above,

Hence,

### Example Question #8 : Residue Theory

Find the residue of the function

.

**Possible Answers:**

**Correct answer:**

Observe

The coefficient of is .

Thus,

.

### Example Question #4 : Residue Theory

Find the residue at of

.

**Possible Answers:**

**Correct answer:**

Let .

Observe,

The coefficient of is since there is no term in the sum.

Thus,